Modeling and Optimization

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Presentation transcript:

Modeling and Optimization Section 4.4b

Do Now: #2 on p.214 What is the largest possible area for a right triangle whose hypotenuse is 5 cm long, and what are its dimensions? 5 Domain: Derivative:

Do Now: #2 on p.214 What is the largest possible area for a right triangle whose hypotenuse is 5 cm long, and what are its dimensions? Critical Point: for for This critical point corresponds to a maximum area!!!

Do Now: #2 on p.214 What is the largest possible area for a right triangle whose hypotenuse is 5 cm long, and what are its dimensions? Solve for y: Solve for A: The largest possible area is , and the dimensions (legs) are by .

More Practice Problems: #18 on p.215 A piece of cardboard measures 10- by 15-in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid. (a) Write a formula for the volume of the box. The base measures in. by in…

More Practice Problems: #18 on p.215 A piece of cardboard measures 10- by 15-in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid. (b) Find the domain and graph of . Graph in by (c) Find the maximum volume graphically. The maximum volume is approximately when

More Practice Problems: #18 on p.215 A piece of cardboard measures 10- by 15-in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid. (d) Confirm this answer analytically. when at our critical point, meaning that this point corresponds to a maximum volume.

More Practice Problems: #36 on p.217 How close does the curve come to the point (3/2, 0)? (Hint: If you minimize the square of the distance, you can avoid square roots.) The square of the distance: Domain:

More Practice Problems: #36 on p.217 How close does the curve come to the point (3/2, 0)? (Hint: If you minimize the square of the distance, you can avoid square roots.) Domain: Minimize analytically: CP: Since changes sign from negative to positive at , the critical point corresponds to a minimum distance. Minimum distance: